Coefficient bounds for starlike functions associated with Gregory coefficients

Theorem 2.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ and $\gamma_1, \gamma_2, \gamma_3$ are given by (2.2). Then we have <span id="page-6-1"></span> $$|H_{2,1}(F_f/2)| := |\gamma_1 \gamma_3 - \gamma_2^2| \le \frac{1}{64}.$$ The inequality is sharp.

Lemma 3.1. ([29]) Let $p \in \mathcal{P}$ be given by (2.1). Then $$|c_2 - vc_1^2| \le \begin{cases} -4v + 2 & v < 0, \\ 2 & 0 \le v \le 1, \\ 4v - 2 & v > 1. \end{cases}$$ For v < 0 or v > 1, the equality holds if and only if $$h(z) = \frac{1+z}{1-z}$$ or one of its rotations. If 0 < v < 1, then the equality is true if and only if $$h(z) = \frac{1+z^2}{1-z^2}$$ or one of its rotations. We obtain the following sharp inequality, which is the upper bound of $|a_3 - \mu a_2^2|$ for the class $\mathcal{S}_G^*$ .

Theorem 3.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ . Then we have $$|a_3 - \mu a_2^2| \le \begin{cases} \frac{1}{12} (1 - 3\mu), & \mu < -\frac{2}{3}, \\ \frac{1}{4}, & -\frac{2}{3} \le \mu \le \frac{4}{3}, \\ \frac{1}{12} (1 - 3\mu), & \mu > \frac{4}{3}. \end{cases}$$ The inequalities are sharp.

Lemma 4.1. ([35]) Let $p \in \mathcal{P}$ be given by (2.1). If 0 < a < 1, 0 < b < 1 and $8a(1-a)\{(b\beta-2\lambda)^2+(b(a+b)-\beta)^2\}+b(1-b)(\beta-2ab)^2 \le 4ab^2(1-a)(1-b)^2$ then $$|\lambda c_1^4 + ac_2^2 + 2bc_1c_3 - \frac{3}{2}\beta c_1^2c_2 - c_4| \le 2.$$ Now, we obtain the following result i.e., a sharp bound for the Zalcman functional in the case where n = 3 for the class $\mathcal{S}_G^*$ .

Theorem 4.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ . Then we have <span id="page-12-1"></span> $$|a_3^2 - a_5| \le \frac{1}{8}.$$ The inequality is sharp.

Lemma 4.2. ([2]) Let $p \in \mathcal{P}$ be given by (2.1) with $0 \le B \le 1$ and $B(2B-1) \le D \le B$ . Then $$|c_3 - 2Bc_1c_2 + Dc_1^3| \le 2.$$ We obtain the following result concerning the sharp bound for the generalized Zalcman functional $J_{2,3}$ for the class $\mathcal{S}_G^*$ .

Theorem 4.2. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_G^*$ . Then we have $$|a_2 a_3 - a_4| \le \frac{1}{6}.$$ The inequality is sharp.

Definition 1.1. Let f and g be two analytic functions in $\mathbb{D}$ . Then f is said to be subordinate to g, written as $f \prec g$ or $f(z) \prec g(z)$ , if there exists a function ²⁰²⁰ Mathematics Subject Classification. Primary 30C45; Secondary 30C50, 30C80. Key words and phrases. Univalent functions, Starlike functions, Gregory coefficients, Hankel determinants, Logarithmic coefficients, Zalcman functional, Fekete-Szego inequality. $\omega$ , analytic in $\mathbb{D}$ with w(0) = 0, |w(z)| < 1 such that f(z) = g(w(z)) for $z \in \mathbb{D}$ . Moreover, if g is univalent in $\mathbb{D}$ and f(0) = g(0), then $f(\mathbb{D}) \subseteq g(\mathbb{D})$ . In [29], Ma and Minda gave a unified presentation of various subclasses of starlike and convex functions by replacing the subordinate function (1+z)/(1-z) by a more general analytic function $\varphi$ with positive real part and normalized by the conditions $\varphi(0) = 1$ , $\varphi'(0) > 0$ and $\varphi$ maps $\mathbb D$ onto univalently a starlike region with respect to 1 and symmetric with respect to the real axis. They have introduced a general class that envelopes several well-known classes as special cases $$\mathcal{S}^*[\varphi] = \{ f \in \mathcal{A} : zf'(z)/f(z) \prec \varphi(z) \}.$$ In the literature, functions belonging to the class $\mathcal{S}^[\varphi]$ are known as the Ma-Minda starlike functions. For $-1 \leq B < A \leq 1$ , the class $\mathcal{S}^[(1+Az)/(1+Bz)] := \mathcal{S}^[A, B]$ is called the class of Janowski starlike functions, introduced by Janowski in [17]. The class $\mathcal{S}^[\beta]$ of starlike functions of order $\beta$ , where $0 \leq \beta < 0$ , is defined by taking $\varphi(z) = (1 + (1 - 2\beta)z)/(1 - z)$ . Note that $\mathcal{S}^ = \mathcal{S}^[0]$ is the classical class of starlike functions. By taking $$\varphi(z) = 1 + \frac{2}{\pi^2} \left( \log \left( (1 + \sqrt{z}) / (1 - \sqrt{z}) \right) \right)^2,$$ we obtain the class $\mathcal{S}^*[\varphi] = \mathcal{S}_p$ of parabolic starlike functions, introduced in [39]. Recently, the coefficient problem and many other geometric properties for class $\mathcal{S}^*[\varphi]$ are studied extensively in [9, 16, 18, 38]. For example, in [9], Deniz has studied sharp coefficient problem for the function $$\varphi(z) = e^{z + \frac{\lambda}{2}z^2} \quad (z \in \mathbb{C}, \lambda \ge 1),$$ which is a generated function of generalized telephone numbers. In [31], Mendiratta et al. obtained the structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for the exponential function $\varphi(z) = e^z$ . It is worth pointing out that, the case when $\varphi$ defined by $$\varphi(z) = \frac{2}{1 + e^{-z}}$$ is a modified sigmoid function which maps the unit disk $\mathbb D$ onto the domain $$\Delta_{SG} = \left\{ z \in \mathbb{C} : \left| \log \left( \frac{z}{2-z} \right) \right| < 1 \right\}.$$ Sharp coefficient problem is studied for this function by Riza et al. in [38], Goel and Kumar in [14]. In [18], Kazımoglu et al. have considered the function $\varphi$ for which $\varphi(\mathbb{D})$ is starlike with respect to 1 and whose coefficients is the Gregory coefficients. Gregory coefficients are decreasing rational numbers of the form $1/2, 1/12, 1/24, 19/720, \ldots$ , which are similar in their role to the Bernoulli numbers and appear in a large number of problems, especially in those related to the numerical analysis and to the number theory. They first appeared in the works of the Scottish mathematician James Gregory in 1671 and have since been rediscovered numerous times. Among the renowned mathematicians who rediscovered them are Laplace, Mascheroni, Fontana, Bessel, Clausen, Hermite, Pearson, and Fisher. Furthermore, Gregory's coefficients can rightfully be considered among the most frequently rediscovered entities in mathematics, with the most recent rediscovery dating back to our century. This frequent rediscovery has led to various names for them in literature, such as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, Cauchy numbers, and more. Consequently, their authorship is often attributed to various mathematicians. In [5,33], the authors have considered the generating function of the Gregory coefficients $G_n$ as follows: $$\frac{z}{\ln(1+z)} = \sum_{n=0}^{\infty} G_n z^n \text{ for } z \in \mathbb{D}.$$ For first few positive integers, the reader may check that $G_0 = 1$ , $G_1 = \frac{1}{2}$ ,  FIGURE 1. The image $\Psi(\mathbb{D})$ which is starlike with respect to 1. $G_2=-\frac{1}{12},~G_3=\frac{1}{24},~G_4=-\frac{19}{720},~G_5=\frac{3}{160}~{\rm and}~G_6=-\frac{863}{60480}.$ In [18], Kazımoglu et al. have considered the function $\Psi(z):=\frac{z}{\ln(1+z)}$ with its domain of definition as the open unit disk $\mathbb D$ . With this function, we define the class $$S_G^* := \{f: f \in \mathcal{A} \ \text{ and } \ zf'(z)/f(z) \prec \Psi(z)\}.$$ Finding the upper bound for coefficients has been one of the central topics of research in geometric function theory as it gives several properties of functions. The main hurdle lies in finding a suitable function from the class that convincingly exhibits the sharpness of the bound. Keeping in mind such hurdle, we investigate the sharp bound of several problems in geometric function theory such as finding sharp bound of Hankel determinant of logarithmic coefficients, sharpness of the Fekete-Szeg¨o inequality, Zalcman functionals. In the subsequent sections, we discuss our findings and background study of individuals.